Curvature Definition and 872 Threads

obviously the equations of tidal forces and geodesic deviation are very similar to lead one to motivate yourself to explore gravity as not a field but as a curved geometry, Einstein also said that if each accelerated frame is locally an inertial one the euclidean geometry of Lorentz can not...

Yes, general relativity is out of my depths for now. Now I've often seen drawings of a gravitational source represented by a dimple ( downward ) on a surface. Yet GR never speaks of a fifth dimension. Nor have I ever seen a dimple upwards that I would suppose would represent repulsive gravity...

I'm reviewing for my final and there is a question I can't seem to solve. If anyone could help me with it I would appreciate it very much. A ruled surface has the parameterization of the form: x(s,t) = A(s) + tB(s) where A(s) is unit speed, |B(s)| = 1. Show that: K<or= to 0. So...

I'm new to physics but very curious about it. I'm 18, and probably will also include physics as my second major in college. A topic that has always bothered me is the curvature of space. If space is curved, due to the planets and stars, then why don't the rays of the sun curve around the...

Superclusters and Voids -- same curvature? According to the mainstream 'standard model', is the geometric curvature of space believed to be exactly the same within superclusters as it is in within voids? In other words, does the much higher gravitational density within a supercluster...

So this book I have (Mathematical Methods for Physics and Engineering, Riley, Hobson, Bence) defines curvature as being: \kappa = \left | \frac{d \hat{\textbf{t}}}{d s} \right | = \left | \frac{d^2 \hat{\textbf{r}}}{d s^2} \right | where t hat is the unit tangent to the curve and r hat...

Hey, hopefully this is a suitable forum to put this in! I've been having a bit of trouble trying to find an example to learn from about finding the raidus of curvature in a simply supported beam. I've got a point loads and a UDL to take into account and seem to find it near impossible to find...

Hey, hopefully this is a suitable forum to put this in! I've been having a bit of trouble trying to find an example to learn from about finding the raidus of curvature in a simply supported beam. I've got a point loads and a UDL to take into account and seem to find it near impossible to find...

Would it be possible to adjust for the curvature of space between 2 points and so by taking the shortcut (a true straight line) beat a light source in a race between the 2 points whilst traveling at less than light speed?

I don't really get GR. Why should curved space and time be a model for gravity? To me, curved space means a observers no longer measure distances as sqrt(x^2+y^2+z^2), but rather, given an x-ordinate, y-ordinate and z-ordinate, the length of the shortest path to that coordinate can be calculated...

I'm having trouble finding the point of the maximum curvature of the line with parametric equations of: x = 5cos(t), and y = 3sin(t). I know the curvature "k" is given by the eq.: k = |v X a|/ v^3 Where v is the derivative of the position vector r = <5cos(t), 3sin(t) > , a is the...

In a response by Pervect to another topic, he mentioned a device called a Forward mass detector, named after its inventor Dr. Robert Forward. It's an intersting device with the claim that it can detect small gradients in the curvature of spacetime. I couldn't find any info regarding...

suppose a curve is not uniformly curved and i would like to describe how "curvy" a segment of this curve is? how would i do to this? i imagine i can take the second derivative and find the average of it over the entire segment and the closer the average is to zero the straight the segment is but...

Greetings all. This is my first post. I'm a newbie to general relativity, but I think I'm getting the hang of it thanks to some helpful professors at UC Berkeley. From what I understand, and now fully believe, there are no external forces applied to an object that is free falling in...

That famous experiment during a solar eclipse, which showed the curvature of light from a star as the light rays passed by the sun, pretty much I gather confirmed Einstein's space time curvature theory. Question: Does spacetime curve into one of those hidden spatial dimensions that M-theory...

Due to mass Space time curves. Consider the case of sun & earth. Sun,since it is heavy will curve the space more than earth.isn't it? Then how come the sun and Earth are being pulled towards each other with same force? The Earth has to straighten the curve (caused by sun)first,and then has...

Hey, I've been a little confused on the concept of gravitons. I know that they are the messenger particle of the gravitational force, but I thought that gravity was a result of the warping of the fabric of spacetime. If a large star warps spacetime, therefore attracting things around it, then...

Homework Statement A temperature controller, designed to work in a steam environment, involves a bimetallic strip constructed of brass and steel, connected at their ends by rivets. Each of the metals is t thick. At 20 degrees C, the strip is L0 long and straight. Find the radius of curvature...

Inside a spherical cavity centered at the Earth's centre, the space-time curvature is 0 or =/= 0? I know Newtonian gravitational field is omogeneously 0, so no field variation, but does GR give a different answer?

Energy-mom tensor does not determine curvature tensor uniquely ? If the energy momentum tensor is known, that fixes the Einstein tensor uniquely from the Einstein eqs. Einstein tensor is built from Riemann contractions so it doesn't fix Riemann uniquely. Does that mean a single energy momentum...

I do wonder if space-time curvature can be applied to artificial satellites ... I think yes because that could be the reason why they are revolving around earth. Doubt:But what happens if they gain velocity more than the escape velocity. I could be conceptually wrong but if the above...

I calculate trace-free Ricci scalars (Phi00, Phi01,Phi02, etc) and scalar of curvature (Lambda=R/24) in Newman-Penrose formalism using a computer package. How can I find the Ricci scalar out of them? I though R was the Ricci scalar but Lambda comes non-zero for a spacetime whose Ricci scalar is...

hallow everyone i am a tenth-grade student in Taiwan.What i want to know is that how to proove the curvature at point (a,(f(a))(assume f(x) is smooth at this point) is f"(a)/(1+f'(a)^2)^(3/2)) i've thought this way:consider a circle first in this circle the curvature at point P is lim...

How do you define curvature for curves on three-dimensional surfaces when the surface is given in the form z=f(x,y)? The resulting formula should be a lot simpler than the one for parametric curves of the form r(t)=(x(t),y(t),z(t)), like it becomes for two-dimensional curves given by y=f(x)...

I will be writing my final exam tomorrow evening, and I am currently terribly stuck on the following practice problems. I have posted my thoughts below each problem. They look tricky to me. It would be very nice if someone could help me out and I will remain eternally grateful for your help...

Homework Statement I need to show that the mean curvature H at p \in S given by H = \frac{1}{\pi} \cdot \int_{0}^{\pi} k_n{\theta} d \theta where k_n{\theta} is the mean curvature at p along a direction makin an angle theta with a fixed direction. Homework Equations I know...

This equation gives us (delta (rho))/rho (which I understand is the fractional perturbation in the energy density), at the time of "horizon entry" (which I'm unsure about). Does this mean the time that decoupling occured?

Can the laplacian of a scalar field be throught of as its curvature (either approximately or exactly)?

Homework Statement Suppose \alpha is a regular curve in \mathbb{R}^3 with arc-length parametrization such that the torsion \tau(s)\neq 0, and suppose that there is a vector Y\in \mathbb{R}^3 such that <\alpha',Y>=A for some constant A. Show that \frac{k(s)}{\tau(s)}=B for some constant B...

This may be a simple question for some of you but it has baffled me for a long time. When we say that spacetime is curved, do we mean that from a flat space of a higher dimension, our spacetime would appear curved, in the same way that the surface of a sphere looks curved when viewed from...

Consider a photon emitted at space-time event E1 and absorbed at space-time event E2 in curved space-time. Since the arc length of the worldline between both events is 0 how can we, with validity, claim that such a path is curved in space-time? Does it not seem to be more correct to claim...

Hi folks. I am a mathematician and my research is on the curvature equation D(\gamma) = F where \gamma is a Lie-algebra valued one-form and F is a Lie-algebra-valued 2-form. I want a very rough idea how fiber bundles and associated vector bundles are used in physics. I've tried...

Is it accurate to claim that space-time curvature in general relativity means a curvature of a space-time with a Minkowski pseudo-metric?

Ok, so I don't know much about general relativity or quantum mechanics but, if gravity effects everything in the universe, and if Heisenberg's uncertainty principle makes it so that you can not have truly empty space (so every point of space has to have some sort of particle occupying it, bc if...

Homework Statement Find the unit speed curve alpha(s) with k(s)=1/(1+s^2) and tau defined as 0. Homework Equations Use the Frenet-Serret equations K(s) is the curvature and tau is the torsion T= tangent vector field (1st derivative of alpha vector) N= Normal vector field (T'/k(s))...

Homework Statement Find the curvature of y = x³ Homework Equations k(x) = \frac{f"(x)}{[1+(f'(x))²]^{3/2} The Attempt at a Solution k(x) = \frac{6x}{(1+9x^4)^{3/2} I got the answer numerically, but I am looking for an explanation of the graph itself. I chose a relatively easy...

EDIT: Is the Schwarzschild coordinate a radius of curvature in the geodesic? And also, in physics, what do I make of a negative radius of curvature?

We can all see what curvature of space looks like, just by throwing a ball and watching it follow the natural geodesic. But what does curvature of time look like? How do we experience it? We typically experience the passage of time in what seems to be a forward linear manner. The...

What is Quantum Gravity and the Curvature of Spacetime and how is it all relevant to one another?

I understand why it is so desirable to be able to write all the laws of physics by the same rule in any system of coordinates. I also nearly understand that the equivalence principle leads to the need of curved spacetime. But how to make that as obvious as possible? Thanks, Michel

A real object is placed at the zero end of a meterstick. A large concave mirror at the 100 cm end of the meterstuck forms an image of the object at the 82.4 cm position. A small convex mirror placed at the 20 cm pisition form a final image at the 6.3 cm point. What is the radius of curvature of...

Could anyone share insights/results/references on hypersurfaces with vanishing extrinsic curvature? In particular, I would be interested in results related to existence (do they always exist, if not when do they exist?) and procedures for constructing them from the background geometry.

I've been stuck on this for ages and would appreciate help on how to do it: On a train, the magnitude of the acceleration experienced by the passengers is limited to 0.050g.If the train is going round a curve at a speed of 220km/hr what's the smallest radius of curvature that the curve can...

Einsteins field equations are nonlinear. One could interpret this to mean that curvature is itself the source of curvature (thus not only mass). Would it be possible to find a stationary (non-zero) solution of the (non-linearised) field equations without a mass being present - a kind of...

General relativity says that the gravitational "field" is just the warping of space by mass. I like to think of the ball on the trampoline analogy. Is dark energy, basically negative pressure, be caused by the natural curvature of spacetime? http://www.geocities.com/ixi_dima_ixi/gr.JPG

What does it mean for something to have an infinite curvature (like a black hole?)?

I was wondering, how does light bend in very intense gravitational fields, if it has no weight? And does anyone have a good source for facts on spacetime curvature, gravity and such? Thanks

Can anyone point me to good reference that fully develops the geometry of geodesic curvature? Most of the ones I have manage to derive it, then show it's the normal to the curve, then never mention it again. I want to know how it relates to the metric, first second or third. Thanks.

Okay, so I was asked to find all the things listed in the topic title given the equation: r(t)=(cos^{3}t)\vec{i} + (sin^{3}t)\vec{j} Now this is a lot of work, especially when it comes to finding the torsion \tau = - \frac{d \vec{B}}{ds} \cdot \vec{N} a total of four derivitives. Maybe I am...

Hi, In two dimensions I am under the impression that the ricci tensor or the scalar curvature equals the negative of the fundamental tensor and the sectional curvature (K). I'd have written it out with the proper symbols but I am new to this forum and this isn't at least a complex question...